Short-term Wave Forecasting as a Univariate Time Series Problem

Transcription

1 Short-term Wave Forecasting as a Univariate Time Series Problem Francesco Fusco December 9 Research Report: EE/9/3/JVR Department of Electronic Engineering, NUI Maynooth Maynooth, Co. Kildare, Ireland

2 Contents The univariate approach to wave forecasting Data analysis 3. Wave spectra Non-stationarity and the Wavelet transform Linearity analysis Predictability measure Choice of cut-off frequency Choice of the sampling frequency Methodology 3. Getting the observations of the signal to predict Forecasting models Cyclical models Auto Regressive (AR) models Sinusoidal extrapolation and the Extended Kalman Filter Neural networks Confidence intervals Results 3 4. Cyclical models Auto-Regressive models Choice of model order Static AR models Adaptive AR models Sinusoidal extrapolation through Extended Kalman Filter (EKF) Neural networks Effects of sampling frequency on prediction Further possibilities Gaussian Processes Auto Regressive Moving Average (ARMA) models Conclusion 59

3 The univariate approach to wave forecasting This report aims to collect and present the main results and conclusions due to the work done by the author on the problem of forecasting the ocean wave elevation at a specific point of the sea surface based on past observations collected at the same point. The problem is strictly connected to the real time control, on a wave by wave basis, of a Wave Energy Converter (WEC), which can, potentially, significantly increase its efficiency and energy capture ability, but that is well known to be described by non-causal relationships in the time domain [],[],[3]. The wave elevation η(k), observed at a particular point with a certain sampling frequency, is treated as a pure univariate time series, so that the forecasting problem consists of determining the prediction ˆη(k + l/k) a number of steps, l, ahead based on all the information up to the current sampling instant k. All the solutions proposed so far in the literature deal with the problem by trying to reconstruct the wave field at a certain point of the sea surface based on one or more distant measurements [4],[5],[6],[7], as shown in figure (b). This approach, however, requires complex numerical models and a large enough array of measurements in order to properly deal with the non-linearities of wave propagation, including wave refraction and multidirectionality. Alternatively, the solution discussed here, based only on local measurements of the wave elevation (or of any other related quantity of interest, such as the wave excitation force), as illusrtrated in figure (a), allows for certain significant advantages: multidirectionality and all the associated complications need not to be considered; the wave propagation laws do not need to be modelled and no simplifying assumptions (e.g. linearity, dispersion relationship) are then required about them; if the considered point corresponds to the position of the WEC, the radiated waves do not affect the measurements; all the well established theory about univariate time series forecasting may be exploited; no additional instrumentation around the device is required (cheap solution). Its validity, however, is limited by the possibility to effectively estimate the wave elevation at the point where the device is located (this problem and some related implications are briefly discussed in section 3.). The available data and its detailed analysis through different tools is presented in section. Then the actual solution to produce the predictions is proposed in the methodology of section 3 and evaluated on real wave data in the results section 4. Some other possible forecasting models, for which results were not produced, and motivations for their unsuitability are discussed in section 5. Conclusions are finally presented in section 6.

4 incoming waves incoming waves observed wave observed wave predicted wave WEC predicted wave WEC (a) Prediction based only on local singlepoint measurements (b) Prediction based on reconstruction of wave field from array of distant measurements Figure : Two main approaches to wave forecasting Data analysis The data available comes from different locations. The Irish Marine Institute provided real observations from a data buoy located in Galway Bay, on the West Coast of Ireland (at approximately 53 o 3 N, 9 o 8 W ). These observations consist of minute records sets for each hour, collected at a sampling frequency of.56 Hz, for parts of years 7 and 8. The location is sheltered from the Atlantic Ocean so that the wave height magnitude is generally small, which makes it not an ideal site for full size WECs, though a wave energy test site has been established there for /4-scale prototypes. Wave elevation time series are also available from the Atlantic Ocean at the Pico Island, in the Azores archipelago, at approximately 38 o 33 N, 8 o 34 W. They are collected in the form of two contiguous 3 minute record sets for each hour, with a sampling frequency of.8 Hz (that is 34 samples for each set).. Wave spectra The main tool for a first analysis of waves is their spectral distribution, the wave spectrum, which shows how the energy is distributed across the different frequency components of the wave, assumed to be completely independent of each other. Although offering limited time-averaged information (a Wavelet transform would offer a more complete information in the time domain, refer to section.) it is still very valuable in order to provide some overall characteristics of the sea conditions in different situations and at different locations. A first analysis, which is interesting to carry out over the available hourly data sets, concerns the distribution of the significant wave height and the peak and mean radian frequency of the spectrum, respectively ω peak and ω mean, and to assess if their behaviors are correlated to each other in some way. The significant wave height is a measure of the mean energy contained in the wave, while ω peak and ω mean can be a way to represent where the spectrum (and so the energy) of the wave is more concentrated. From figure, it is clear how high energy wave systems present a much narrower spread of ω peak and ω mean, centered at a low frequency (about rad/s for the Galway Bay data, 3

5 even lower for the wave systems from Pico), which means that the energy is more concentrated at the low frequencies and the spectral distribution has a well defined narrow peak (swell). The lower the energy, on the other hand, the more the distance between the peak and the mean frequency, which denotes a much flatter spectrum where the high frequency wind waves have a similar energy content to the low frequency swell. The sample spectra of figure 3 are particularly illustrative in this respect.. Non-stationarity and the Wavelet transform A better and more complete understanding of the energy distribution in the waves can only be be achieved through an analysis in both the frequency domain and the time domain, that is something like a continuous evolution of the wave spectrum over time. An interesting tool that has been successfully applied to this proposal is the Wavelet transform (references [8],[9]), which shows a good resolution both in time and frequency, thus making the application of the shortterm Fourier transform, where only a compromise between time and frequency resolutions can be obtained, less attractive for waves analysis. Wavelets have been successfully implemented in signal and image processing, ordinary and partial differential equation theory, numerical analysis and communication theory [9]. On the other hand, the application of the wavelet transform to ocean engineering and oceanography is not frequent. This is mostly due to the fact that not all such applications provide quantitative results, so that the wavelet transform has been regarded has an interesting tool to produce colorful pictures, yet purely qualitative results [9]. The Wavelet transform of a signal, x(t), is defined from the following expression: W T (t, b) = + x(t)g (t; τ, b)dτ () Here, g (t; τ, b) is the complex conjugate of a continuously translated and dilated mother wavelet function g(t): g(t; τ, b) = ( ) t τ g () b b where t is the translation parameter, corresponding to the position of the wavelet as it is shifted through the signal, b is the scale dilation parameter determining the width of the wavelet. At low frequencies (high value of scale b), the frequency resolution is better but the time resolution is poor (more ambiguity regarding the exact time). On the other hand, at higher frequencies (low scale b), the frequency resolution is poorer and the time resolution is better. This main characteristic of the Wavelet transform, which is due to the fact that the signal is multiplied with a window whose width is changed as the transform is computed for each spectral component [9], is a significant improvement to the Short Term Fourier Transform (STFT), where the window is constant and an appropriate compromise has to be made between time and frequency resolution. In fact, a finer time resolution at higher frequencies is important because the signal is changing faster, while a poorer time resolution at low frequencies can be acceptable because the signal is changing more slowly. 4

6 Hs [m] 7 Hs [m] ωpeak [rad/s] ωmean [rad/s] ωmean [rad/s] (a) Hs [m] 7 Hs [m] ωpeak [rad/s] (b) Figure : Correspondence between significant wave height Hs, peak frequency ωpeak and mean frequency ωmean at the locations: (a) Galway; (b) Pico. 5

7 =.3 m.5 =.34 m.4 S(ω) [m s/rad] S(ω) [m s/rad] ω [rad/s] (a) ω [rad/s] = 6.48 m =.6 m 4.4 S(ω) [m s / rad] 8 6 S(ω) [m s / rad] ω [rad/s] (b) ω [rad/s] Figure 3: Typical high and low energy spectra at the locations: (a) Galway; (b) Pico. 6

8 A correspondence between the scale value b and the Fourier period T can be found, and it depends on the specific mother wavelet, g(t), chosen. In the case of the very common Morlet wavelet, the following expression can be derived [9]: b = c + c + T αt (3) 4π where c is a parameter defining the Morlet mother wavelet, also having the nature of a frequency in some sense. Note that the physical dimension of b is the time (seconds). Expression (3) can then be used to have physical meaning (the frequency) of the scale dimension of the transform W T (t, b), which can then become a W T (t, f) or W T (t, ω) in a very straightforward way. Figures 4 and 5 show the Wavelet transform for some wave elevation data sets, respectively from Galway Bay and from Pico. Although, as mentioned, it is not easy to derive any quantitative result from them, it is still possible to get some interesting information out of them. Note, in fact, how the different frequency components of the Fourier spectrum may appear in different moments so that, in the short term, the bandwidth of the wave signal could actually be much narrower than what the Fourier transform suggests..3 Linearity analysis Ocean waves, like most systems in the real world, are not linear, and it would be helpful and valuable to quantify how far from linearity they are so that, in the particular case of wave forecasting, a proper model can be chosen. Linearity in the case of waves means linear superposition of harmonic components (sines and cosines). The emerging of non-linearities in waves manifests itself, in the first instance, by a non-gaussian distribution of the wave elevation time series around its mean value (zero, the water surface level), due to the presence of higher and narrower peaks than troughs (modelled by quadratic, cubic,... and so on, terms of the linear harmonic components, according to Stokes theory []). The degree of asymmetry depends on the significance of the water depth with respect to the wavelength (the difference between the greatest elevation and the greatest depression is minimum for h >> λ and maximum for shallow water). This nonlinearity is expected, therefore, to be more consistent at the high energy and low frequency components of the waves when the water depth is not sufficiently large. A higher order statistical analysis, through the indices of kurtosis and skewness, would be useful to assess the extent of this non-linearity. Skewness and kurtosis, in fact, can determine the degree of Gaussianity of the distribution of the wave elevation around the mean water surface level, and statistical analysis of the time history of wave records indicates that the wave profiles are normally distributed apart some very small deviations on rare occasions []. Given the nth-order central moments, µ n, of a certain random variable x: µ n = E{(x µ) n } (4) where E{ } is the expectation operator and µ is the mean E{x} of the random variable x, then the two indices of kurtosis, κ, and skewness, γ, are given by 7

9 (a) (b) 8 (c) Figure 4: Wavelet transform for three different data sets from Galway Bay

10 (a) (b) Figure 5: Wavelet transform for three different data sets from the Pico Island. 9

11 []: γ = µ 3 σ 3 (5) κ = µ 4 σ 4 (6) where σ µ is the nd-order central moment. The skewness measures the asymmetry of the distribution: γ = denotes a symmetric distribution, otherwise, if γ >, the distribution is more concentrated around a value greater than the mean and viceversa if γ <. The kurtosis represents a degree of peakedness as compared to a Gaussian distribution, in which case κ = 3: if κ > 3 the distribution is termed leptokurtic (sharp peak), otherwise, if κ < 3, it is termed platykurtic (mild peak) []. Figures 6 and 7 show this analysis for the two locations of Galway Bay and the Pico Island. As expected, higher energy data sets show a slight deviation of the indices of skewness and kurtosis from the normality condition, particularly in the case of the Galway Bay, where the water depth is smaller (nearly m), while in the Pico Island only very high energy waves move away from normality, as shown in the detailed wave distribution of figure 7(c). From a wave energy point of view, although the interest is obviously focused on high energy waves, the non-symmetry effect may not be an issue if deep water locations are considered. There is, however, another possible non-linearity, which unfortunately is less quantifiable and can only be analysed through visual inspection. This is due to the interactions occurring between different harmonic components of the wave system, which are neglected in the classical linear wave theory and in the Fourier-Wavelet analysis. A higher order analysis through the Bispectrum (refer to Ochi []) revealed to be quite effective in order to detect these interactions; however, as said, a real quantification would be hard to carry out and probably not really significant. This non-linearity is known to be more present in wind waves, which represent high frequency and low energies wave systems and are less interesting from a wave energy point of view. A low-pass filtering of the wave elevation time series, in particular, may help to reduce their effect so that they should not be taken into account by the forecasting model. Figures 8 and 9 represent the bispectra calculated for some significant data sets from the Galway Bay and the Pico island, respectively. The off-diagonal components appear if an interaction between the two corresponding frequencies exists, and it is evident how they usually are strong for high frequency wind waves interacting with swell (as in figure 8(c)) or for broad spectra resulting from the superposition of different wave systems (figures 8(a) and 9(b)). The bispectrum is, on the other hand, much more concentrated around the diagonal for narrow banded swell systems, as can be noted from figures 8(b) and 9(a)..4 Predictability measure As the focus of this study is on the multi-step-ahead prediction of the wave elevation (or of any connected quantity), one of the striking questions is this: Is there any chance to predict future values of a given signal? Usually, we design a predictor for a special signal or problem and then measure the resulting prediction quality. If there is no a priori knowledge on the optimal predictor, the achieved prediction gain will depend strongly on the particular prediction

12 [m] [m] kurtosis (a) Distribution of skewness and kurtosis skewness. =.3 m skewness =.7 kurtosis = 3.9. =.3 m skewness =.6 kurtosis = probability. probability..5.5 wave elevation [m] wave elevation [m] (b) Probability distribution of two sample wave elevation data sets Figure 6: Galway bay Gaussianity analysis

13 [m] [m] kurtosis (a) Distribution of skewness and kurtosis skewness.. =.39 skewness =.4 kurtosis = 3.3 =.64 skewness =.65 kurtosis = probability. probability wave elevation [m] wave elevation [m] (b) Probability distribution of two sample wave elevation data sets. = 6.48 m skewness =.43 kurtosis = probability wave elevation [m] (c) Probability distribution of a very high energy wave elevation data set Figure 7: Pico Island Gaussianity analysis

14 .8 S(ω) [m s/rad] ω [rad/s] ω [rad/s] ω [rad/s] (a).3 S(ω) [m s/rad] ω [rad/s] x ω [rad/s] ω [rad/s] (b).5 S(ω) [m s/rad]..5 ω [rad/s] ω [rad/s] 3 x ω [rad/s] (c) Figure 8: Bispectrum of some data sets in Galway Bay

15 S(ω) [m s/rad] ω [rad/s] ω [rad/s] ω [rad/s] (a) S(ω) [m s/rad] ω [rad/s] 3 x ω [rad/s] ω [rad/s] (b) Probability distribution of two sample wave elevation data sets Figure 9: Bispectrum of some data sets in Pico Island 4

16 model used. Here, it is argued that, for prediction feasibility analysis, it is not necessary to design any predictors; we just have to know how much information about future signal values can be obtained from the past [3]. A simpler measure of predictability than the very general approach proposed in the literature (based on the mutual information notion [3],[4],[5],[6]) will be adopted here, which supposes that a linear relationship exists that relates the future values of the wave elevation to the past. This is, of course, a limiting assumption but it is still very valuable to provide at least a qualitative study over the predictability of the wave elevation. In particular, a predictability index R (l) is estimated, defined as the ratio of the variance of the optimal l-step-ahead prediction, ˆη(k+l/k), to the variance to the real wave elevation, η(k): R (l) E{ˆη(k + l/k) } E{η(k) } = ˆσ l E{η(k) } (7) where it is supposed that the wave elevation η(k) has a zero mean and the optimal l-step-ahead prediction error variance is defined as ˆσ l E{ê(k+l/k) }. A very efficient algorithm for the estimation of R(l), under the assumption of a linear univariate time series, was proposed in [7] and it is adopted here for the analysis of the available wave data. Figure shows the estimated predictability index R (l), for a forecasting horizon of 5 samples of different wave systems at the two locations of Galway Bay and Pico Island. As expected from any real-world time series, it is a non-increasing function of the prediction horizon. All the wave systems considered for the Galway Bay location, figure (a), show a relatively poor predictability, which dies out very quickly after 4 seconds (5 samples), with a slightly better behavior of the narrow banded (although low energy) wave system and of the high energy one. A much better predictability (index R is relatively high for more than 5 seconds) results for high energy and narrow banded wave systems at Pico Island, figure (b), mostly due to the smaller influence of the non-linearities analysed in section.3, consisting of either asymmetry in the wave distribution or non-linear interactions between different frequency components. In a wave energy context, however, one might be interested in forecasting only the high energy components, so that a low-pass filter can be applied to the time series and a focus would be put exclusively on the low frequency components. In figure, the estimated predictability index R(l) is shown for the prefiltered wave systems at Galway Bay and when different cut-off frequencies, ω c, are applied. It is clear, by comparison with figure, how the overall predictability significantly improves with respect to the non-filtered waves. Moreover, the smaller the cut-off frequency, i.e. the lower the frequencies we limit the analysis to, the better the predictability of the time series, when a swell at the low frequencies is present, and more accurate predictions, further in the future, can be expected. Figure (c), in fact, referring to the wind waves system of figure 3(a), shows only a significant improvement for the highest cutoff frequency considered, ω c = rad/s. This shall, however, not be a big issue in a wave energy context, when these wave systems are not of interest for their very low energy content and their high frequency content (as compared to lower frequency dynamics of wave energy converters). 5

17 .6 m, wide banded swell.3 m, narrow banded small swell.34 m, mostly wind waves.8 R estimate (a) Galway Bay data m, wide banded mixed swell and wind waves.39 m, narrow banded swell 6.48 m, wide banded big swell.7.6 R estimate (b) Pico Island data Figure : Predictability indices R(l) energy estimated for datasets with different 6

18 Figure depicts the situation for the Pico Island wave elevation data sets. In this case, the narrow banded swell and the low energy mixed wave systems of figures (a) and (b) derive the best improvement from considering only the low frequency swell. The same improvement is not achieved for the wide banded swell system of figure (c), whose spectrum can be seen in figure 3(b), and is also more strongly affected by the non-linearities analysed in section.3, in particular the non-gaussianity. Note that this analysis does not depend on any actual forecasting technique that might be implemented, so that the prediction accuracy will also depend on the chosen method, but some sort of upper bound for the attainable accuracy, irrespective of the utilised algorithm, is set here (although limited to the range of the possible linear forecasting models)..5 Choice of cut-off frequency From the analysis carried out in the previous sections, particularly in. and.4, it emerged how low frequency components are the most interesting from a wave energy point of view and, at the same time, with respect to high frequency wave components, have a more regular behavior so that they are predictable more accurately and further into the future. It was stated, therefore, that one might focus the forecasting algorithm exclusively on the low frequency components, completely neglecting the rest of the signal. If it is considered, however, that the prediction shall be utilised by a controller in order to improve the WEC ability to extract energy from the waves, we may expect that the energy contained in the frequency components neglected by the forecasting procedure represents a loss of extracted energy. That is, the controller is not able to improve the system response to those frequency component not considered in the prediction algorithm. Note that very high frequencies may be lost anyway due to the lowpass filtering dynamics of the WEC device itself but, in general, this depends on its operating principle and its design parameters. The choice of the cut-off frequency of the prediction system can therefore be seen as a compromise between the accuracy improvement in the forecasts (which should improve the energy extraction of the WEC) and the loss of the energy carried by higher frequency components of the incident wave. If the exact relationship between extracted energy and prediction accuracy was known, then a cost functional quantifying the compromise may be calculated and an optimal cut-off frequency may be found. Of course, such a function depends on so many variables (kind of device, control architecture, sea state, etc...) that it would be very hard and, at the same time, not really worth getting a real and complete model of it. A rough quantification of this cost functional may, however, be carried out at this stage by making no assumptions neither on the forecasting algorithm nor on the device. It can be still a very valuable approach, in the author s opinion, as it would not require any significant effort or accurate knowledge of the problem. It can, moreover, be easily extended and become more and more accurate as new pieces of information are known about the overall problem and are then included in the cost functional. For the moment, it will be supposed that the energy extracted by a general non-specified device equals the energy contained in the forecasted wave weighted by the accuracy of the prediction. The accuracy 7

19 .8 =.3 m ω c = rad/s ω c =.5 rad/s ω c =. rad/s ω c = rad/s R estimate (a).8 =.3 m ω c = rad/s ω c =.5 rad/s ω c =. rad/s ω c = rad/s R estimate (b).8 =.34 m ω c = rad/s ω c =.5 rad/s ω c =. rad/s ω c = rad/s R estimate (c) Figure : Predictability of some wave elevation data sets from Galway Bay when low-pass filtering with different cut-off frequencies ω c is applied. 8

20 .8 =.39 m ω c = rad/s ω c =.5 rad/s ω c =. rad/s ω c = rad/s R estimate (a) =.3 m ω c = rad/s ω c =.5 rad/s ω c =. rad/s ω c = rad/s.8 R estimate (b).8 = 6.48 m ω c = rad/s ω c =.5 rad/s ω c =. rad/s ω c = rad/s R estimate (c) Figure : Predictability of some wave elevation data sets from Pico Island when low-pass filtering with different cut-off frequencies ω c is applied. 9

21 wave predictability % ( seconds ahead) wave energy considered % energy that is possible to extract 8 energy % ω c [rad/s] Figure 3: Energy extracted by the Wave energy converters thanks to the wave prediction and dependant on the cutoff frequency of the prediction, here, is substituted with the achievable accuracy as given by a predictability index such as the one of equation (7). The theoretical absorbed energy, E a, is then calculated as: E a = P r (E tot E neg ) (8) where E neg is the energy neglected by the prediction algorithm, E tot is total energy in the waves and P r is a measure of the predictability. In figure 3 it is clear how the cut-off frequency is a compromise between prediction accuracy and energy cutoff, and if the functional (8) would be exact or reasonably accurate, it can be chosen at the maximum of the E a curve. The quantitative results here presented are not really significant because of the many simplifying (and also unrealistic) assumptions. It is, however, an interesting approach to the choice of the appropriate cut-off frequency when other parts of the problem will be better understood (utility of the prediction for the WEC performance, prediction error influence on the control, etc...)..6 Choice of the sampling frequency In general, if the spectrum of a signal has a limited support [ ω m ], then all the information is maintained if the signal is discretised with any sampling radian frequency ω s ω m /π. Lower sampling frequencies give raise to the aliasing phenomenon, thus causing the sampled time series not to be uniquely representative of the original signal. If a wave elevation time series is low-pass filtered before the prediction, this means that it can be sampled without any loss of information with ω s ω c /, where ω c is the cut-off frequency of the filter (assuming an ideal filter with instantaneous transition). Intuitively, a certain time span of the wave elevation signal is represented by fewer samples if the sampling frequency is lower. It

22 R estimate Pico Island,.5 m, ω cut =.7rad/s Nyquist freq: ω cut /π.8 Hz f s =.8 Hz f s =.64 Hz f s =.47 Hz f s =.3 Hz f s =.56 Hz Figure 4: Estimated predictability for a wave elevation time series from Pico Island, when different sampling frequencies are adopted. may be, therefore, that the choice of the sampling frequency could affect the performance of a prediction algorithm, in particular its forecasting horizon in terms of seconds. In theory, the information that the past of a signal has about its future, will not be affected by the sampling frequency, if aliasing is avoided. Therefore, a proper forecasting model that manages to extract all the information to produce the prediction, should not perform differently by changing the sampling frequency. Figure 4 shows, in fact, how the predictability of a certain wave elevation data set is not affected by a change in the sampling frequency, when this is greater than the Nyquist frequency. In practice, however, when performing the prediction, some differences might arise, so it is interesting to assess the effect of the sampling frequency on the prediction accuracy of the forecasting models that will be presented in section 3. This analysis will be carried out in section Methodology Here a range of possible forecasting models is presented. Firstly, in section 3., some considerations are given about how the wave elevation can be actually measured at the same point of the sea where a device is located. Then, the models are presented in section 3., and a methodology to derive proper confidence intervals along with the predictions is proposed in section Getting the observations of the signal to predict The prediction of the incident wave (or any related physical quantity) on a WEC, based only on its past history, presents the main issue of getting the actual measurement of the signal being predicted. In the case of the incident wave elevation, there is no direct access to it at the actual point where the device is located (the device being an oscillating body or an oscillating water column). There is, however, a concrete possibility of estimating the variable of interest from the measurements of other more accessible variables. The accuracy of this estimation will, of course, depend on the accuracy of the mathematical/numerical model relating the measures to the signal to be deduced. Some

23 considerations relevant to this problem will be provided here for a general oscillating body in single mode of motion. A basic approximation of the equations of motion of a WEC is given by the following expression: [m + m( )] ẍ + t k(t τ)ẋ(τ)dτ + Sx = f e (t) + f ext (x, ẋ, t) (9) where x(t) is the displacement of the body along the considered degree of freedom, m and m( ) are the mass and added mass of the body, k(t) is the impulse response function relating the oscillation velocity of the body to its radiation force, S is the buoyancy coefficient, f e (t) is the wave excitation force and f ext (x, ẋ, t) represents any external, for example provided by the power take-off system.. If the measurements of the body motion are available, together with an accurate model of the system, the excitation force f e (t) can directly be estimated from equation (9). The records of this estimate, ˆf e (t), can then be utilised to provide predictions of its future behavior. Alternatively, the excitation force can be used to derive the actual incident wave elevation and the forecasting problem could be focused on the latter. Note, however, that the excitation force and the wave elevation, η(t), are related by a noncausal relationship: f e (t) = + h(t)η(t τ)dτ () with h(t) for t <, so predictions of the wave elevation are also required to estimate the current excitation force ˆf e (t). Consider also that it is the wave excitation force that is actually required in order to compute an optimal reference for the optimal control of the system.. If the total wave force f w (t) is measured instead, by means for example of pressure transducers on the body surface, then the excitation force could be determined through the following expression: f e (t) = f w (t) + t k(t τ)ẋ(τ)dτ + Sx f ext (x, ẋ, t) () which still needs the motion measurements in order to calculate the radiation and the buoyancy forces. Note that equation () directly derives from (9) if we consider that the total force f w (t) [m + m( )] ẍ. For the oscillating bodies, therefore, it seems reasonable to focus on the measurement and prediction of the wave excitation force, rather than the incident wave elevation. In this view, the approach followed in this report to forecast the wave elevation is, however, still valuable, particularly in view of the lowpass filtering applied to the signal prior to the prediction. The wave excitation force, in fact, is nothing else than the wave elevation filtered by the dynamics of the body (which obviously has a lowpass characteristics).

24 3. Forecasting models 3.. Cyclical models From linear wave theory [], a real ocean sea state may be modelled as a linear superposition of waves with different frequencies and propagating in different directions: η(x, y, t) = + dω +π π A(ω, β) cos(ωt kx cos β ky sin β + ϕ i (ω))dβ () where k is the wave number and β represents the direction of propagation in the x-y plane. If a specific location (x, y ) is considered, the following simplified expression can then be obtained: η(x, y, t) = + dω +π π A(ω, β) cos(ωt + φ(ω, β)) (3) where the directionality information is obviously lost and the constant terms kx cos β and ky sin β are included in the phase φ(ω, β). From this knowledge about the real process it is quite straightforward to choose, as a forecasting model for the wave elevation, a simple cyclical model, where the frequency domain is of course discretised [8],[9]: η(t) = m a i cos(ω i t) + b i sin(ω i t) + ζ(t) (4) i= An error ζ(t) has been introduced and the phase and amplitude information for each harmonic component is now contained in the parameters a i and b i. The model (4) is completely characterised by the parameters a i,b i and by the frequencies ω i. It could then be fitted to the data through some non-linear estimation procedure (the model is non-linear in the frequencies) and utilised to predict the future behavior of the wave elevation time series. It needs, however, to be adapted to the time variations of the wave spectrum (amplitudes and phases of the frequency components are non-constant), so that a first approach [9] has been considered, where the frequencies are chosen in the model design phase and then kept constant during its utilisation and estimation. In this way the model becomes perfectly linear in the parameters a i,b i and can be easily estimated and on-line adapted to the spectral variations of the sea. The problem of choosing the frequencies can be divided in two sub-problems:. Choice of the range: This is a quite easy matter, as statistical information about the location can be utilised to properly define an upper and lower bound for the range. At this point, one may decide to include the range of higher frequencies where the low energy wind waves are, or to simply consider a narrower range including only the swell.. Distribution of the frequencies in the range: A robust choice would be a constant spacing between the frequencies over all the range, but a more 3

25 efficient non-homogeneous distribution was also proposed in [9]. The latter however suffers from the problem of specificity, so that if the wave spectrum changes the frequencies might not be appropriate any more. If the frequencies are kept constant, then it would not be a proper choice. Once the frequencies are determined, a model for the amplitudes has to be chosen. In [8], [9] it was pointed out how they have to be adaptive to the wave, as constant amplitudes gave very poor results. Two adaptive models are proposed here, in particular: Structural model: based on Harvey s structural model [], the model (4) is expressed in the following discrete time form: η(k) = m ψ i (k) + ζ(k) (5) i= [ ] ψi (k + ) ψi = (k + ) [ cos(ωi T s ) sin(ω i T s ) sin(ω i T s ) cos(ω i T s ) ] [ ] ψi (k) ψi (k) + [ ] wi (k) wi (k), i =,... m (6) where it can be verified that ψ i () = a i and ψ i () = b i. From equation (6), then, the following state space form, which is more familiar to work with, is easily derived: where x(k + ) = Ax(k) + w(k) η(k) = Cx(k) + ζ(k) (7) x(k) [ψ (k) ψ (k)... ψ m (k) ψ m(k)] T R m (8) w(k) [w (k) w(k)... w m (k) wm(k)] T R m (9) {[ ]} cos(ωi T A diag s ) sin(ω i T s ) R m m () sin(ω i T s ) cos(ω i T s ) C [... ] R m () Dynamic Harmonic Regression (DHR): Introduced by Young [], it expresses a cyclical model of the type of eq. (4), where the a i and b i parameters evolve according to a Generalised Random Walk: [ ] [ ] [ ] [ ] [ ] xi (k + ) α β xi (k) δ ɛi (k) x i (k + ) = γ x i (k) + ɛ i (k) () x i = a i for i =,... m x i m = b i for i = m +,... m where x i models a slope for the evolution of each parameter x i. The disturbance terms ɛ i and ɛ i are still assumed to be Gaussian noises and introduce the variability in the model. A particular form of () was implemented in this study where the dynamic matrices are chosen in order to represent Harvey s local linear trend []: [ ] [ ] [ ] [ ] [ ] xi (k + ) xi (k) ɛi (k) x i (k + ) = x i (k) + ɛ i (k) (3) 4

26 for i =,,... m. A state space form, then, can easily be derived, resulting in the following model: x(k + ) = Ax(k) + ɛ(k) η(k) = C(k)x(k) + ζ(k) (4) where x(k) [x (k) x (k)... x m (k) x m(k)] T R 4m (5) ɛ(k) [ɛ (k) ɛ (k)... ɛ m (k) ɛ m(k)] T R 4m (6) {[ ]} A diag R 4m 4m (7) C(k) [cos(ω T s )... cos(ω m T s ) sin(ω T s )... sin(ω m T s ) ] R 4m (8) Both the models have the advantage of a state space representation, which is particularly suited to the application of the Kalman filter for a recursive on-line adaption. The initialisation is provided through means of regular least squares on a number of past observations and then the Kalman filter is applied online, once a proper covariance matrix for the state and output disturbances is provided. When the estimate of the model s parameters, ˆx(k/k), is available at any instant k, the l-steps-ahead prediction ˆη(k + l/k), based on the information up to k, is obtained through the free evolution of the model: ˆη(k + l/k) = C(k + l)a lˆx(k/k) (9) There are, however, some strong limitations to this approach with cyclical models[8],[9], that will be highlighted also in the results, section 4: The use of constant frequencies requires, for the sake of robustness, a dense and complete set, which adds considerable complexity to the model, and It is not clear how to choose the covariance matrices for the Kalman filter implementation In the next section 3.., it will be shown how AR models implicitly overcome these difficulties in a very effective, and simple, way. 3.. Auto Regressive (AR) models As a pure time series problem is under study, there is the advantage of the existence of a well established theory, from the time series field, which it is possible to utilise. In a comparison with the cyclical models, where the a priori knowledge that we have about the real system is explicitly taken into account, it is particularly interesting to analyse the properties of classical AR models. The wave elevation η(k) is supposed to be linearly dependent on a number n of its past values: n η(k) = a i η(k i) + ζ(k) (3) i= 5

27 where a disturbance term ζ(k) has been included. If the parameters a i are estimated and the noise is supposed to be Gaussian and white, the best prediction of the future wave elevation ˆη(k + l/k) at instant k is then given by: n ˆη(k + l/k) = â i (k)ˆη(k + l i/k) (3) i= where, obviously, ˆη(k + l i/k) η(k) if k + l i k (i.e. the information is already acquired and there is no need for prediction). The properties of such a very simple forecasting model become clearer if an explicit solution of the difference equation (3) is provided []: n ˆη(k + l/k) = b i (k)f i (l) (3) Here, the coefficients b i (k) depend only on the forecasting origin (so they stay constant at each instant for the complete prediction time horizon) and are a function of the initial conditions (the past n observations), whereas f i (l) are functions of the lead time l and, in general, they include damped exponential and damped sinusoidal terms completely determined by the roots p i of the transfer function ϕ(z) describing equation (3) in the Z-domain: i= η(z) = ζ(z) ϕ(z) ζ(z) n i= (z p i) (33) The general shape of the prediction function is therefore completely determined by the poles, p i, while the particular realisation of this general structure is determined, at each sampling instant, by the past values of the time series. It is particularly interesting to analyse the shape of the forecasting function (3) in the case of m/ (when m is even) couples of complex-conjugate poles, p i and p i : m/ ˆη(k + l/k) = c i (k) p i l sin( p i k + ϕ i (k)) (34) i= Thus, an AR model with only complex-conjugate poles is implicitly a cyclical model, where the frequencies are related to the phase, p i, of each pole and the amplitudes and phases of the harmonic components are related to the last n observations of each time instant k, so that they adapt to the observations. Note, then, that an adaptivity mechanism is already present even if the AR model parameters are only estimated once on a batch data set. A classical estimation approach is to minimise the squared errors sum, which has a linear solution provided by regular least squares. Only the frequencies, in this case, are fixed, while amplitudes and phases are automatically updated on the basis of the recent past information. A further degree of adaptivity can be introduced with an on-line estimation of the AR model parameters, a i, which would introduce an on-line adaptivity of the frequencies as well. If we express equation (3) in vectorial form: η(k) = ψ(k) T ϑ(k) + ζ(k), (35) where ψ(k) [η(k ) η(k )... η(k n)] T R n (36) ϑ(k) [a (k) a (k)... a n (k)] T R n, (37) 6

28 then a general recursive estimation of the time-varying parameters vector, ϑ(k), is given as: ˆϑ(k + ) = ˆϑ(k) ] + K(k) [η(k + ) ψ(k) T ˆϑ(k) (38) as a combination of free evolution and innovation. How the gain vector K(k) R n is chosen depends on which specific estimation algorithm is utilised. The most common approaches are the recursive least squares with forgetting factor and the Kalman Filter, which are outlined in the following subsections: Recursive Least Squares (RLS): With this approach the following functional is minimised: k J(ϑ(k)) = λ k j [η(j) ψ(j)ϑ(k)], (39) j= where more weight, via the forgetting factor λ <, is given to recent observations according to an exponential law. The recursive algorithm for the optimal value of ϑ(k) that minimises the functional J(ϑ(k)) respects the general form of equation (38), with the gain K(k) given as: K(k) = P (k)ψ(k) (4) P (k) = P (k ) λ + ψ T (k)p (k )ψ(k) The forgetting factor λ is typically chosen in the range [.97,.995]. (4) The matrix P (k) R n n represents the covariance matrix of the estimate ˆϑ(k): P (k) E{ ˆϑ(k) ˆϑ T (k)} (4) It is interesting to introduce also the information matrix R(k) R n n, defined as: R(k) ψ(k)ψ(k) T +λψ(k )ψ(k ) T +...+λ k ψ()ψ() T +R() =..., which can also be shown to correspond to [3]:... = λr(k ) + ψ(k)ψ(k) T (43) R(k) P (k) (44) One main problem of recursive least squares with a forgetting factor is that, if the measurements do not add new information to the system, that is ψ(k) is approximately zero for a certain time. Therefore, the information matrix decreases until it can get close to the null matrix (or only some of its eigenvalues tend to zero). A the same time, dome of the elements of the corresponding gain K(k) may significant increase. When ψ(k), then, increases in magnitude, the estimate ˆϑ(k) can experience a very large growth, known as the phenomenon of blow-up. Some regularisation solutions have been proposed to cope with this problem [4]. One possibility is to monitor the matrix P (k) (R(k)), and reset its values to acceptable ones when its eigenvalues assume too large (small) values [3]. Another approach may be a variable forgetting factor, based on the state of the process (steady or transient) [3], or regularisation of the information matrix R(k) [5]. 7

29 Kalman Fitler: If the evolution of the state vector ϑ(k) is assumed to be a random walk: ϑ(k) = ϑ(k ) + ε(k) (45) where ε(k) is a Gaussian random process, then the Kalman Filter may be applied, resulting in the recursive form of (38), where the gain K(k) is now given by [6]: K(k) = Q(k)ψ(k) (46) with Q(k) = P (k ) R + ψ T (k)p (k )ψ(k) P (k) = P (k ) + R P (k )ψ(k)ψ(k)t P (k ) R + ψ T (k)p (k )ψ(k) (47) (48) where R (k) = E{ε(k)ε(k) T } and R = E{ζ(k) }. Here, P (k) still represents the covariance matrix of the estimate ˆϑ(k) Sinusoidal extrapolation and the Extended Kalman Filter The main problem with the cyclical model is the choice of the frequencies, that must be kept constants in order for the model to be linear in the parameters. Efficient linear algorithms for the recursive estimation of the cycles amplitudes (and phases) can therefore be exploited. This means that the accuracy of the model is strictly connected to its capacity to cover as much as possible of the typical range of frequencies where the wave systems at the considered location are mostly concentrated. This approach lacks of efficiency and requires a much more complex method then what would actually be required. In reality few regular waves in the typical range will be active at each time instant. A more intelligent solution, then, would be to consider a few (or even one) cyclical component with an adaptive frequency which is updated on-line with the real observations. We propose, therefore, to model the wave elevation as a single cyclical component as in the Harvey structural model, equation (6), but with a time-varying frequency ω(k): ] [ ] ψ(k + ) ψ = (k + ) [ cos(ω(k)ts ) sin(ω(k)t s ) sin(ω(k)t s ) cos(ω(k)t s ) η(k) = ψ(k) + ζ(k) ] [ ψ(k) ψ (k) + [ ] ε(k) ε (k) (49) where ε(k), ε (k) and ζ(k) are random disturbances and η(k) is the wave elevation. Along with the components ψ(k) and ψ (k), the frequency ω(k) also needs to be estimated. A state vector x(k), composed of the quantities that need to be estimated, is then defined as: x(k) [ψ(k) ψ (k) ω(k)] T R 3 (5) 8

30 The system (49) can then be redefined in terms of x(k) as: ψ(k + ) cos(ω(k)t s ) sin(ω(k)t s ) ψ(k) ε(k) ψ (k + ) = sin(ω(k)t s ) cos(ω(k)t s ) ψ (k) + ε (k) ω(k + ) ω(k) κ(k) η(k) = ψ(k) + ζ(k) (5) In (5) a model for the adaptivity of the frequency ω(k) has been introduced, where a simple random walk is proposed, driven by the additional white noise κ(k). The model, of course, is non-linear in the frequency and an explicit state space structure cannot be formulated, so the following form is adopted: { x(k + ) = f(x(k), w(k)) (5) η(k) = [ ] x(k) + ζ(k) where f(x( ), w( )) R 3 is a vectorial non linear function and w(k) [ε(k) ε (k) κ(k)] T R 3 is the vectorial form of the state disturbance. The optimal estimate (in the sense of minimising the variance) for x(k), based on the observations Z k = {η(), η(k )}, is known to be given by the conditional mean value: ˆx(k/k ) = E{x(k)/Z k } (53) An explicit expression which is usable in practise is difficult to obtain, though, because of the non-linearity of the system. It is well known, however, that a very efficient algorithm for linear (and Gaussian) systems can be adopted, which is the Kalman filter. However, an application of the latter to non-linear problems can also be found in literature, known as the Extended Kalman Filter (EKF). The EKF assumes that the discrete time steps (T s in our case) are sufficiently small to permit the prediction equations to be approximated by a linearised form, based on the truncation of the Taylor expansion of the model (5) at the first order [7]: [ ] d x(k + ) f(x(k), w) + f(x(k), w(k)) (x(k) x(k)) (54) dx x(k)=x(k) where x(k) represents an opportune working point for the state of the system, whose variation is supposed to be small within the time step T s considered. Based on this approximation, the following state space form can be derived for the model (5): { x(k + ) A(k)x(k) + F (k) + w(k) (55) η(k) = Cx(k) + ζ(k) where A(k) [ ] d f(x(k), w(k)) R 3 3 (56) dx x(k)=x(k) F (k) f(x(k), w) A(k)x(k)R 3 (57) C [ ] R 3 (58) 9

31 The iterative equations for computing the gain K(k) of the Kalman filter [6] can directly be derived from this linear state space form (55), where the operating point at each step is set as the optimal one step-ahead estimate of the state,x(k) = ˆx(k + /k), which is updated based on the non linear equation: ˆx(k + /k) = f(ˆx(k/k), [ ] T ) (59) ˆx(k + /k + ) = ˆx(k + /k) + K(k) [η(k + ) C ˆx(k + /k)] (6) The direct extension to an n-frequencies model may be given by the following: x (k + ) f (x (k), w (k)) x(k) =... =... x n (k + ) f n (x n (k), w n (k)) (6) η(k) = [... ] x(k) + ζ(k) As will be shown in the results of section 4.3, this model has the problem that the Kalman filter would update all the frequencies in the same way, which makes it not really effective Neural networks It was shown in section.3 how the non-linearities appearing in the big low frequency waves, due to the relatively small water depth, are not really relevant. It is however interesting, in the authors opinion, looking at a comparison of the other models with a most widespread tool for time series modelling and forecasting such as neural networks. In spite of either the great modelling capability and the easiness of building up a suitable structure, neural networks have the great disadvantage of offering a model completely enclosed in a black box where any analysis and properties evaluation is prevented. So, while in the cyclical and AR models an analysis of the estimated parameters and frequencies and their variations in an adaptive structure can provide indications about the real process behavior and its main characteristics, this would not be possible with neural networks. For the problem under study, a non-linear relationship of the following type is created through a multilayer perceptron [8]: η(k) = NN(η(k ), η(k ),... η(k n)) (6) so that the dependance between the current wave elevation and n past values is realised. The model is then trained through the back propagation algorithm on a set of batch data and utilised for multi-step-ahead prediction. This is, of course, not the only possibility and many others could be considered. For example, a priori knowledge about the process (which would always be a more appropriate approach) may be included and a non-linear relationship of the following type may be considered instead: η(k) = NN(cos(ω T s k + ϕ ),... cos(ω n T s k + ϕ n )) (63) but some of the limitations outlined in section 3.., when cyclical models where considered, due to an appropriate choice of the frequencies, are still present. In section 4.4 results will be shown and compared with the cyclical and AR models, for different neural network topologies, with two hidden layers and different numbers of inputs (regression ). 3